# 8 3 Study Guide and Intervention Special Right Triangles

## What is a Special Right Triangle? A special right triangle is a right triangle with side lengths that are in a particular ratio. There are three types of special right triangles: 30°-60°-90° triangles, 45°-45°-90° triangles, and the 3-4-5 right triangle. These triangles have special features that make them easier to work with in solving problems. For example, the side lengths of a 45°-45°-90° triangle can be found by multiplying the length of the shorter leg by the square root of two.

## Using the Pythagorean Theorem to Solve Problems The Pythagorean Theorem is a mathematical rule that states that the sum of the squares of the two shorter sides of a right triangle is equal to the square of the longest side. This theorem is used for solving problems involving special right triangles. By plugging in the length of the two shorter sides of the triangle, the length of the longest side can be determined. This can be done for all types of special right triangles.

## Solving for Unknown Side Lengths One of the most common uses of special right triangles is to solve for unknown side lengths. By using the properties of special right triangles, you can determine the length of the missing side. For instance, if you know the length of the shorter leg of a 30°-60°-90° triangle, you can use the special ratio to calculate the lengths of the other two sides. You can also use the Pythagorean Theorem to determine the length of the longest side.

## Using Special Right Triangles to Find Angles Special right triangles can also be used to find angles. By plugging in the known side lengths of a triangle into the Pythagorean Theorem, you can find the angle measures of the triangle. For instance, if you know the length of the two shorter sides of a 45°-45°-90° triangle, you can use the Pythagorean Theorem to determine the length of the longest side. From there, you can find the angles of the triangle by dividing the length of the longest side by the length of the shorter sides.

## Conclusion Special right triangles can be used to solve many types of problems. By understanding the properties of these triangles, you can find side lengths and angles with ease. Using the Pythagorean Theorem and the special ratios of the sides, you can find unknown side lengths and angles in special right triangles. Special right triangles are an invaluable tool in mathematics.